1. Field of the Invention
The present invention relates to a method and an apparatus for evaluating motion characteristics of a gear based on tooth profile deflection of the gear.
2. Discussion of the Prior Art
One example of a method of obtaining an amount of tooth profile deflection of a gear in the direction perpendicular to each tooth surface is disclosed in an article, pages 345-354, "GEARS", vol. 2, 6th edition, Oct. 1, 1966, Nikkan Kogyo Shinbunsha, Tokyo. The method disclosed in this article will be briefly described referring to FIG. 5.
An involute spur gear to be evaluated (hereinafter referred to as "workpiece gear") is fixed to a base circle disk 10 such that the workpiece gear and the disk 10 are coaxial with each other. The disk 10 has a diameter which is equal to that of an ideal base circle of the workpiece gear. A straightedge bar 20 having a flat side surface or straightedge 18 is supported so as to be freely movable substantially along the flat side surface 18, such that the flat side surface 18 is held in rolling contact with a cylindrical surface of the base circle disk 10. On the straightedge bar 20, there is supported a pivotal arm 23 having a contact element in the form of a contact ball 24 such that the arm 23 is pivotable about its fixed end, so that the contact element 24 at the free end of the arm 23 is freely movable in a direction substantially parallel to the flat surface 18, as long as the contact element 24 is located near the flat surface 18, that is, as long as the angle of pivotal movement of the arm 23 is sufficiently small. The base circle disk 10 and the straightedge bar 20 are arranged so that the the straightedge bar 20 is movable with its flat side surface 18 in rolling contact with the cylindrical surface of the disk 10, without slipping of the flat surface 18 on the cylindrical surface of the disk 10. During a rotary movement of the workpiece gear together with the disk 10, the the ball 24 is kept in contact with a tooth surface of the workpiece gear, whereby the contact ball 24 describes or follows a tooth profile curve 26. If the workpiece gear had no tooth profile deflection, the tooth profile curve 26 would be an involute profile. If the workpiece gear has a certain amount of tooth profile deflection, the path of movement of the contact ball 24 deviates away from the involute curve in the direction of movement of the straightedge bar 20. Accordingly, the amount of deviation of the contact ball 24 from the involute curve represents the amount of the tooth profile deflection of the workpiece gear.
Conventionally, the motion characteristics (e.g., vibration characteristic and dynamic load characteristic) of the workpiece gear are evaluated based on the tooth profile deflection obtained as described above. However, since there is not a definite relationship between the amount of tooth profile deflection and the motion characteristics of the workpiece gear, the evaluation of the motion characteristics based on the obtained tooth profile deflection inevitably depends on experience and guesswork of the evaluator. Consequently, the accuracy of the evaluation is limited.
In view of the situation in the prior art discussed above, the inventor has made a research in an effort to find out parameters by which the motion characteristics of the workpiece gear can be accurately evaluated. As a result, the inventor discovered a fact that an amount of variation in the load (hereinafter referred to as "load variation") which acts on the surface of each tooth of a pair of gears during rotation thereof in mesh with each other is largely affected by an error in the base circle radius (hereinafter referred to as "base circle radius error") of the workpiece gear. The base circle radius error is interpreted to mean an amount of deviation of the radius of the instantaneous base circle (which will be described) from the radius of the base circle as generally defined in the field of gear generation or gear geometry. There will be described a relationship between the load variation and the base circle radius error, which has been found on a pair of involute helical gears rotating in a single-flank meshing fashion, by way of example.
To begin with, it is assumed that one of the pair of involute helical gears which is positively driven is a driving gear having no tooth profile deflection, while the other gear is a driven gear which is driven by the driving gear and which has a certain amount of tooth profile deflection. In other words, it is assumed that the driven gear has a resultant value of the actual tooth profile deflections of the driving and driven gears. Further, the pair of involute helical gears are converted into an equivalent pair of involute spur gears, which has the same rotary motion as the helical gear pair. The spur gear equivalent to the driving involute helical gear having no tooth profile deflection is named a first gear, while the other spur gear equivalent to the driven involute helical gear having tooth profile deflection is named a second gear. The present inventor found that the load variation Fd(.theta..sub.2) can be represented by the following equation: EQU Fd(.theta..sub.2)=M.multidot.[T.sub.2 .multidot..DELTA.Rb(.theta..sub.2)/J.sub.2 -.DELTA.Rb'(.theta..sub.2).multidot..omega..sub.20 .sup.2 ]
where,
M=J.sub.1 .multidot.J.sub.2 /[Rb.sub.10.sup.2 .multidot.[i.sup.2 .multidot.J.sub.1 +J.sub.2 ]], PA0 J.sub.1 :moment of inertia of the first gear, PA0 J.sub.2 :moment of inertia of the second gear, PA0 Rb.sub.10 :radius of ideal base circle of the first gear, PA0 i:gear ratio, PA0 T.sub.2 :torque transmitted to the second gear, PA0 .theta..sub.2 :angle of rotation of the second gear when the first gear is rotated by .theta..sub.1, PA0 .DELTA.Rb(.theta..sub.2):base circle radius error of the second gear, PA0 .DELTA.Rb'(.theta..sub.2):differentiated value of .DELTA.Rb(.theta..sub.2) obtained by differentiating .DELTA.Rb(.theta..sub.2) by the angle .theta..sub.2 of rotation of the second gear, PA0 .omega..sub.20 :mean angular velocity of the second gear.
The process in which the above equation was obtained is described in detail in the inventor's report entitled "Rotational vibration of a helical gear pair with modified tooth surfaces", which was delivered at the 69 th national conference held on Oct. 16, 1991 by the Japan Society of Mechanical Engineers. The report is included in "TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS", Nov. 1991, Vol. 57, No. 543.
In essence, the load variation Fd of an involute helical gear largely depends on the base circle radius error .DELTA.Rb(.theta..sub.2) when the involute helical gear is rotating at a relatively low velocity under a relatively large load, and largely on the differentiated value .DELTA.Rb'(.theta..sub.2) when the gear is rotating at a relatively high speed under a relatively small load. Therefore, the motion characteristics of the involute helical gear can be accurately evaluated by obtaining the base circle radius error .DELTA.Rb(.theta..sub.2) and its differentiated value .DELTA.Rb'(.theta..sub.2).